Our proof combines ideas from the two long lines of work on -completness (e.g. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}) and -completeness (e.g. [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}). It also uses ideas from our recent paper on the communication complexity of Nash equilibrium in potential and congestion games [20]}.
